Problem: Solve for $q$, $ -\dfrac{1}{3q - 3} = -\dfrac{10}{15q - 15} - \dfrac{q - 5}{3q - 3} $
Solution: First we need to find a common denominator for all the expressions. This means finding the least common multiple of $3q - 3$ $15q - 15$ and $3q - 3$ The common denominator is $15q - 15$ To get $15q - 15$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ -\dfrac{1}{3q - 3} \times \dfrac{5}{5} = -\dfrac{5}{15q - 15} $ The denominator of the second term is already $15q - 15$ , so we don't need to change it. To get $15q - 15$ in the denominator of the third term, multiply it by $\frac{5}{5}$ $ -\dfrac{q - 5}{3q - 3} \times \dfrac{5}{5} = -\dfrac{5q - 25}{15q - 15} $ This give us: $ -\dfrac{5}{15q - 15} = -\dfrac{10}{15q - 15} - \dfrac{5q - 25}{15q - 15} $ If we multiply both sides of the equation by $15q - 15$ , we get: $ -5 = -10 - 5q + 25$ $ -5 = -5q + 15$ $ -20 = -5q $ $ q = 4$